Integrable Lagrangian correspondences and the factorization of matrix polynomials

M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 38–49, April–June, 1991.     

Non-Crossing Tableaux

In combinatorics there is a well-known duality between non-nesting and non-crossing objects. In algebra there are many objects which are standard, for example, standard Young tableaux, standard monomials, and standard bitableaux. We adopt a point of view that these standard objects are really non-nesting, and we find their non-crossing counterparts.     

Minimal Bar Tableaux

Motivated by recent results of Stanley, we generalize the rank of a partition λ to the rank of a shifted partition S(λ). We show that the number of bars required in a minimal bar tableau of S(λ) is max(o, e + (ℓ(λ) mod 2)), where o and e are the number of odd and even rows of λ. As a consequence we show that the irreducible projective characters of S_n vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur’s Q_λ symmetric functions in terms of the power sum symmetric functions. Received November 20, 2003     

Tableaux and noncommutative Schubert polynomials

Laboratoire Informatique Théorique et Programmation, Université Paris VII, Paris, France. Published in Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 3, pp. 63–64, July–September, 1989.     

Cyclic Tableaux and Symmetric Functions

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p_n in terms of the elementary symmetric function e_n and the homogeneous symmetric function h_n. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.     

Factorization and Poisson correspondences

The Darboux transformation as an example of an integrable infinite-dimensional Poisson correspondence is discussed in the context of the general factorization problem. Generalizations related to energy-dependent Schrödinger operators and to Kac-Moody algebras are considered. We also present the finite-dimensional reductions of the Darboux transformation to stationary flows.Dedicated to the memory of Ira DorfmanTranslated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 225–245, November, 1995.     

Clone and hypervariety correspondences

The general correspondence between clones and hypervarieties was set out by Walter Taylor in [5], By mapping each varietyV to its cloneClone V Taylor obtained inverse isomorphisms between the lattice of all clone varieties and the lattice of all hypervarieties. Taylor also described a more specific correspondence in [6], using the 1-clone Cl₁ V of a varietyV. Since 1-clones are just monoids, he found a one-to-one correspondence from the uncountably infinite lattice of monoid varieties into the lattice of all hypervarieties. In this paper we show how Taylor's construction may be carried out forn-clones, for anyn 1. Thus we usen-clones to restrict the general correspondence to a family (_n) of correspondences, including the monoid correspondence as the special casen=1. These correspondences pick out certain families of hypervarieties, then-closed hypervarieties (forn 1), and we show that there are at least countably many such hypervarieties for eachn. This represents some progress towards the goal of understanding the structure of the lattice of all hypervarieties.Presented by W. Taylor.Research supported by NSERC of Canada.     

Fiber-wise birational correspondences

We prove that each fiber-wise birational correspondence between smooth fibrations into (Fano) hypersurfaces, which is biregular on a generic fiber, is a fiber-wise isomorphism. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 120–130, July, 2000.     

Random correspondences and nonlinear equations

The question of existence of random solutions of nonlinear equations involving random correspondences is studied by direct use of the results from the deterministic case without resorting to iterative techniques. Applications to stochastic control and nonlinear random operator equations are given.     

Patterns in Shi Tableaux and Dyck Paths

Order - Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type A, the set $\mathcal...     

Commuting algebraic correspondences and groups

In this paper, the naturalness of the appearance of the group approach in the study of problems of algebraic commutation is considered. New, previously unknown solutions are obtained. Special attention is paid to an example of commutation obtained from the icosahedral equation. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 662–670, May, 1997. Translated by A. I. Shtern     

On Homomorphisms Indexed by Semistandard Tableaux

We describe techniques which may be used to compute the homomorphism space between Specht modules for the Hecke algebras of type A. We prove a q-analogue of a result of Fayers and Martin and show how it may be applied to construct homomorphisms between Specht modules. In particular, we show that in certain cases the dimension of the homomorphism space is given by the corank of a matrix whose entries we write down explicitly.     

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux – ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Compositions of theta correspondences

Theta correspondence θ over is established by Howe (J. Amer. Math. Soc. 2 (1989) 535). In He (J. Funct. Anal. 199 (2003) 92), we prove that θ preserves unitarity under certain restrictions, generalizing the result of Li (Invent. Math. 97 (1989) 237). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method (J. Algebra 190 (1997) 518; Representation Theory of Lie Groups, Park City, 1998, pp. 179-238; The Orbit Correspondence for real and complex reductive dual pairs, preprint, 2001), one can compose theta correspondences to obtain unitary representations. We call this process quantum induction.     

On the Orthogonal Tableaux of Koike and Terada

Many different definitions have been given for semistandard odd and even orthogonal tableaux which enumerate the corresponding irreducible orthogonal characters. Weightpreserving bijections have been provided between some of these sets of tableaux (see [3], [8]). We give bijections between the (semistandard) odd orthogonal Koike-Terada tableaux and the odd orthogonal Sundaram-tableaux and between the even orthogonal Koike-Terada tableaux and the even orthogonal King-Welsh tableaux. As well, we define an even version of orthogonal Sundaram tableaux which enumerate the irreducible characters of O(2n).     

Character correspondences and nilpotent fully-ramified sections

The focus of this paper is a well-known lemma: A continuous group homomorphism between profinite groups which has dense image must be an open surjection. Its importance lies in the common occurrence of infinite inversely directed systems of finite groups. A limit, in the category of groups, is rarely useful. Instead, experience reveals that an inverse limit in the topologized sense completely characterizes the system. Principles like the above lemma empower this model.

The author needs “good” inverse limits for a broad clas of categories. We develop an idea from the 1960s: Given a categoryn C, expand it with purely formal projective limits.

Although definition of such a category is elementary, the approach has difficulties. In the past, crucial manipulations refused generalization. Our thesis is that, by tightening hypotheses, the method creates a category valuable to analysis. In this paper, we

1. review and modify the categorical formulation,

2. prove a structure theory for pullbacks and inversely directed systems, and

3. generalize the indicated lemma.     

On Symmetry and Positivity for Domino and Ribbon Tableaux

Inspired by the spin-inversion statistic of Schilling, Shimozono and White and Haglund et al., we relate the symmetry of ribbon functions to a result of van Leeuwen, and also describe the multiplication of a domino function by a Schur function. Received July 11, 2004